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Finite-Wordlength Effects

We have just seen how the FDN can be seen as a simple DWN having a not-necessarily physical scattering matrix. In order to provide an easy control over the decay, the scattering matrix has to satisfy the condition of losslessness (27). A finite-precision implementation of the FDN might incur in limit cycles or overflow oscillations, due to departures from the infinite-precision lossless prototype. Departures can be of two kinds: the finite-precision scattering matrix does not satisfy the lossless condition (27), or the round-off noise in the matrix by vector multiplication ${\bf A}{\bf p}^+$ introduces signal amplitude modifications. By assuming that the scattering matrix satisfies (27) in a large extent even in finite precision, it is possible to apply the arguments used in [27,28,26,6] for the DWNs, in order to avoid limit cycles or overflow oscillations. If the matrix by vector multiplication is performed in the straightforward way as a collection of inner products, and the matrix coefficients have the same $n$ bits of precision as the signals, it is just sufficient to perform these order-$N$ inner products in the extended precision of $2 n + N - 1$ bits, and apply a passive truncation scheme on the output signal. In two's complement arithmetic, a simple passive truncation scheme is the following:

As far as the condition on the losslessness of the scattering matrix is concerned, general requirements for the construction of ``structurally lossless,'' or at least ``structurally passive'' scattering matrices have to be worked out. This topic, previously touched by Gray [6] in the $N=2$ case, will be discussed in a forthcoming paper, since a complete treatment would enlarge the scope of this paper significantly.


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``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in IEEE Transactions on Speech and Audio, vol. 5, no. 1, pp. 51-60, Jan. 1996.

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Copyright © 2005-03-10 by Davide Rocchesso and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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